Abstract We define solitons for the generalized Ricci flow on an exact Courant algebroid. then a family of flows left-invariant Dorfman brackets algebroid over simply connected nilpotent Lie group, generalizing bracket in way that might make this new useful study geometric such as flow. provide explicit examples both constructions Heisenberg group. also discuss solutions to

Given a pair of Lie algebroid structures on a vector bundle A (over M) and its dual A∗, and provided the A∗-module L = (∧A ⊗ ∧T ∗M) 1 2 exists, there exists a canonically defined differential operator D̆ on Γ(∧A ⊗ L ). We prove that the pair (A,A∗) constitutes a Lie bialgebroid if, and only if, D̆ is a Dirac generating operator as defined by Alekseev & Xu [1].

We apply the Atiyah–Singer index theorem and tensor products of elliptic complexes to cohomology transitive Lie algebroids. prove that Euler characteristic a representation algebroid $A$ over compact manifold $M$ vanishes unless $A=TM$, general Künneth formula. As applications, we give short proof vanishing result for principal bundle calculated using invariant differential forms, show certain ...

In this paper some geometric structures on Finsler Lie algeboids are studied and h-basic distinguished connections introduced. Specially, Ichijy? connection that is a special investigated. The generalized Berwald algebroids presented, as particular case of Wagner-Ichijy? connection, studied. Moreover, the Wagner algebroid introduced equivalent conditions for space given. Finally, an optimal con...

In categorified symplectic geometry, one studies the categorified algebraic and geometric structures that naturally arise on manifolds equipped with a closed non-degenerate n + 1-form. The case relevant to classical string theory is when n = 2 and is called ‘2-plectic geometry’. Just as the Poisson bracket makes the smooth functions on a symplectic manifold into a Lie algebra, there is a Lie 2-...

We begin with a short presentation of the basic concepts related to Lie groupoids and Lie algebroids, but the main part of this paper deals with Lie algebroids. A Lie algebroid over a manifold is a vector bundle over that manifold whose properties are very similar to those of a tangent bundle. Its dual bundle has properties very similar to those of a cotangent bundle: in the graded algebra of s...

We propose a new framework for constructing geometric and physical models on nonholonomic manifolds provided both with Clifford – Lie algebroid symmetry and nonlinear connection structure. Explicit parametrizations of generic off–diagonal metrics and linear and nonlinear connections define different types of Finsler, Lagrange and/or Riemann–Cartan spaces. A generalization to spinor fields and D...

We first define the concept of Lie algebroid in convenient setting. In reference to finite dimensional context, we adapt notion prolongation a over fibred manifold manifold. Then show that this construction is stable under projective and direct limits adequate assumptions.